Package jline.api.sn

Class SnToQrfAlpha

java.lang.Object
jline.api.sn.SnToQrfAlpha
public final class SnToQrfAlpha extends Object
Per-station load-dependent rate scaling alpha(i,n) for the QRF bounds.

Port of matlab/src/api/sn/sn_to_qrf_alpha.m.

The load-dependent QRF arms carry a scaling alpha(i,n) that multiplies EVERY rate out of station i while it holds n jobs, completions mu and background phase changes v alike -- see the q construction in Mapqn_qrf_noblo_mmi_ld. That is exactly the rate law of 

   an infinite server        alpha(i,n) = n
   a c-server station        alpha(i,n) = min(n, c_i)
   limited load dependence   alpha(i,n) = sn.lldscaling(i,n)
so the three COMPOSE BY MULTIPLICATION and not one of them is an approximation: the relaxed chain is the model's own, and the QRF answer keeps whatever status it had on a single-server model.

WHERE IT STOPS BEING THE MODEL'S OWN IS PHASE-TYPE SERVICE AT A STATION THAT SERVES SEVERAL JOBS AT ONCE. The QRF local state carries ONE phase per station, a faithful description of one job in service and of nothing else: min(n,c) jobs served in parallel each advance through a phase of their own, and no scaling of a single-phase process reproduces that joint motion. A multiserver or delay station must therefore be exponential -- scaling a PH server by min(n,c) would answer a DIFFERENT chain, so the relaxation would stop containing the model's stationary distribution and the number would bound nothing. Limited load dependence at a SINGLE server is exempt and admits PH freely: one job is in service whatever the rate.

THE UTILIZATION NORMALIZER IS THE DECLARED PEAK, NOT max(alpha). LINE reports U = T*S/peak at every station whose rate scales with the population, one convention shared by multiserver, lld and class dependence. peak is therefore nservers(i) times the largest lld scaling the model can REACH, and not max(alpha(i,:)): at c = 3 with N = 2 the reachable alpha peaks at 2 while the station still has three servers, and normalizing by 2 would report a utilization the model never attains. Infinite at a delay, where LINE reports U = QN instead.

Since:
LINE 3.0

  • Method Details

    • snToQrfAlpha

      public static SnToQrfAlpha.Result snToQrfAlpha(NetworkStruct sn)
      Parameters:
      sn - network structure
      Returns:
      the scaling, or a Result carrying a non-empty msg